Munich Personal RePEc Archive

A Hybrid Approach for Forecasting of

Oil Prices Volatility

Komijani, Akbar and Naderi, Esmaeil and Gandali Alikhani,

Nadiya

University of Tehran, Faculty of Economic, Tehran, Iran, University

of Tehran, Faculty of Economic, Tehran, Iran, Science and Research

Branch, Islamic Azad University, khouzestan-Iran

4 January 2013

Online at https://mpra.ub.uni-muenchen.de/44654/

MPRA Paper No. 44654, posted 06 Mar 2013 07:00 UTC

1

A Hybrid Approach for Forecasting of Oil

Prices Volatility

Abstract

This study aims to introduce an ideal model for forecasting crude oil price volatility.

For this purpose, the ‘predictability’ hypothesis was tested using the variance ratio

test, BDS1 test and the chaos analysis. Structural analyses were also carried out to

identify possible nonlinear patterns in this series. On this basis, Lyapunov exponents

confirmed that the return series of crude oil price is chaotic. Moreover, according to

the findings, the rate of return series has the long memory property rejecting the

efficient market hypothesis and affirming the fractal markets hypothesis. The results

of GPH test verified that both the rate of return and volatility series of crude oil price

have the long memory property. Besides, according to both MSE and RMSE criteria,

wavelet-decomposed data improve the performance of the model significantly.

Therefore, a hybrid model was introduced based on the long memory property which

uses wavelet decomposed data as the most relevant model.

Key Words: Forecasting, Oil Price, Chaos, Wavelet Decomposition, Long Memory.

1 Brock, Dechert and Scheinkman, (1987)

2

1. Introduction

Strong correlation between oil price and political as well as economic situation in the

international scope, on the one hand, and the deep impacts of crude oil price on

country-specific macroeconomic indicators, on the other, have made oil market and

crude oil price fluctuations an ever-hot topic in the energy economics which has

always received great attention from consumers, producers, governments, policy

makers and scholars (Wang et al., 2011). Oil price, also, is identified as a key factor

in financial markets because it significantly affects option pricing, portfolio

management and risk measurement processes (Wei et al., 2010).

Oil importing and exporting countries pay great attention to oil price variations.

However, this issue is of utmost importance for major oil exporting countries like

Iran. For example, in

Iran, shaping 90 percents of country's export value, crude oil and gas exports

constitute approximately 60 percents of government's income (Farzanegan and

Mrakwart, 2011); Also, oil revenue is the major source of government spending and

directly affects fiscal and monetary policies implemented by the government.

Subsequently, oil price is recognized as the main source of macro-level fluctuations

in the oil-based government-driven economy (komijani et al., 2013; Mehrara and

Mohaghegh, 2012; Mehrara and Oskoui, 2007).

Considering its vital importance for decision makers, forecasting major indicators of

oil price market is broadly accepted as both a scientific and applied goal. A brief

review of the literature proves that almost all modeling and forecasting techniques

_from theoretically designed structural models to thoroughly numerical models_

have been implemented in the oil market. Structural models are relatively more

useful in explaining the status of a market; however, in forecasting studies most

financial market analysts tend to use time series models. Such models which are

relatively powerful in practice are based on the Efficient Market Hypothesis (Ozer

and Ertokatli, 2010). As a modern competitive alternative, in recent years, a new

approach has been suggested. This approach known as Fractal Market Hypothesis is

based on Chaos Theory. This theory guides researchers towards detailed

explanations of special events in the market. Technically speaking, implementing

maximal Lyapunov exponent (to verify the predictability of time series by nonlinear

3

models) and its inverse (to determine time span of forecasting) are two crucial

prerequisites. This is a very important step because the results of linear estimation

for those series recognized as chaotic are not valid.

The goals of this paper are to (a) investigate the predictability of the fluctuations of

the time series of Iran’s crude oil price on the basis of Fractal Market Hypothesis and

Chaos Theory; (b) model these fluctuations by long memory models (in particular,

ARFIMA-FIGARCH model); and (c) compare the performance of this model with a

hybrid model consisted of ARFIMA-FIGARCH model and wavelet decomposition.

For the aim of the study, the daily data from 1/2/2002 to 11/3/2011 were applied.

From these 2536 observations, approximately 99 percent were used for estimations

and the remaining 17 observations for out-of-sample forecasting.

1. Iran’s Oil Price: a Brief Historical Review

For numerous political and economic reasons, oil market is always volatile; though

they vary in size and durability. In sum, from 1973 to 2009, these factors had major

impacts on oil price:

1. Arab Oil Embargo (1973-74) 2. Islamic Revolution in Iran (1978-79)

3. Iraq’s Invasion of Iran (1979-80) 4. Saudi Arabia Excess Production1 (1985-86)

5. Iraq’s Invasion of Kuwait (1989-90) 6. Financial Crisis in South Asia (1997-

98)

7. September 11 attacks in New York 8. US Invasion of Iraq (2003-04)

(2001)

9. Some Geopolitical Issues (2006-07) 10. US financial crisis (2007-11)

In 2008, the extensive consequences of the financial crisis caused oil price to reduce

from 150 US $ to nearly 35 US $ per barrel. Since then, in spite of some temporary

decreases, crude oil price has consistently increased in the world market.

Considering continuous increase in demand in the oil market, political instability in

the Middle East, recent threats on the Islamic Republic of Iran, and the deep

financial crisis in the Euro region, it is expected that this increasing trend will

continue in the following years. Figure 1 depicts movements of Iran’s heavy oil price

in the past decade.

1.Due to invention of a new method called Netback

4

Source: EIA1 Reports

Figure 1: Iran’s heavy oil price (2002-2011)

2. Literature Review: Chaos Theory in Oil Market

Chaos refers to a state of utter confusion or disorder. Philosophically, chaos is a total

lack of organization in which accident determines the occurrence of events, but

technically chaos is the order of seemingly disordered systems. Chaos theory studies

such systems. Chaotic systems are deterministic, meaning that their future behavior

is fully determined by their initial conditions, with no random elements involved. In

other words, the deterministic nature of these systems does not make them

predictable. But the analysts who are not aware of the nature of the system (or does

not know it well enough), cannot distinguish between a chaotic and a random

system. Unfortunately statistical tests cannot, also, distinguish between them. So,

considering the measurement accuracy limit, the accuracy of forecasts based on

usual statistical or econometric techniques, continuously _at an exponential rate_

decreases.

Chaotic systems are highly sensitive to initial conditions, an effect which is

popularly referred to as the butterfly effect. Small differences in initial conditions

(such as those due to rounding errors in numerical computation) yield widely

1 Energy Information Administration

5

diverging outcomes for chaotic systems, rendering long-term prediction impossible

in general. Chaotic systems are nonlinear dynamic systems which (1) are highly

sensitive to initial conditions; (2) have unusual complicated absorbents; (3) sudden

structural breaks in their trajectory are distinguishable (Prokhorov, 2008). However,

it should be noted that:

1. The behavior of chaotic systems though seems random, in essence can be

theoretically explained by deterministic rules and equations. Nevertheless, even

though we accept the existence of equations explains the source of their chaotic

behavior, proximity and inaccuracy (though very small) are inevitable due to

measurement limitations.

2. Even very small inaccuracy in initial conditions, because chaotic system is highly

sensitive to them, leads to huge differences between expected and realized values

in the long term. In other words, as time passes, forecasted series and measured

values of realized series totally diverge so much that previous forecasts are no

longer reliable; a fact called unpredictability in the long run (Williams, 2005).

Lyapunov exponent tests and the inverse of maximal Lyapunov exponent allow us to

recognize the dynamics of disturbing term in a chaotic process and distinguish

between a chaotic error term and a stochastic process.

On the other hand, wavelet technique decomposes a non-stationary series into two

(or more) approximation and detail series. Decomposed series can be modeled

independently and separately (Lineesh and John, 2010). Therefore, wavelet analysis

can be useful in describing the signals with discontinuous or fractal structures in the

financial market. It also allows the removal of noise-dependent high frequencies

while conserving the signal bearing high frequency terms (Homayouni and Amiri,

2011). This feature reduces the magnitude of error components and subsequently

leads to more accurate forecasts.

Econometricians explain and forecast the dynamics of highly volatile price series by

Generalized Autoregressive Conditional Heteroskedasticity (GARCH)-Type models.

Such models perform well in tracking two critical features of the data; volatility

clustering and Fat Tail series. These critical features can originate from such factors

as sudden shocks, structural variations, domestic demands, global situations and

political events can be included in these volatility models (Vo, 2011).

Oil market is one of the ever-volatile markets that make forecasting a challenging

task. In addition, since oil is a strategic good, any price movement in oil market

6

immediately affects other financial markets (Erbil, 2011). So, many scholars have

modeled and forecasted oil price fluctuations with GARCH-Type models. But the

crucial question is whether or not using such models is verified. To find appropriate

answer, we need to know whether (i) oil price series predictable. If yes, (ii) whether

oil price series is chaotic? And if yes (iii) whether oil price series has a long-memory

property?. Many studies have been conducted on the predictability of oil price (e.g.,

Arouri et al., 2010a; Alvarez-Ramirez et al., 2010; Charles and Darné, 2009; Alvarez-

Ramirez at al., 2008). Afees et al., 2013; Kang, 2011; vo, 2011; Wei, 2010;

Mohammadi and Su, 2010; Arouri et al., 2010b; Cheong, 2009 also confirmed

volatility of the oil price. Furthermore, the long memory feature was also confirmed

to exist in oil markets in Mostafaei and Sakhabakhsh, 2011; Prado, 2011; Wang et al.,

2011; Choi and Hammoudeh, 2009 study. The present study attempts to combine the

findings of these studies and present a model that can make the most accurate

forecasts about oil price volatility.

4. Methodology

4.1. Diagnosing Chaotic Processes

Several tests have been proposed in the chaos theory literature to distinguish between

stochastic and chaotic processes. Some of these tests recognize stochastic processes

while others are aimed at diagnosing chaotic processes; the former are called indirect

and the latter direct tests. Indirect tests like BDS test almost always consider the

residual series of a linear or nonlinear regression and test whether it is stochastic.

Therefore, when the null hypothesis of randomness of residual is rejected, one

cannot conclude that the series is necessarily chaotic (Brock et al., 1997; Takens,

1981).

Before the formulization of chaos theory, the concept of Lyapunov exponent was

applied to verify the stability of linear or nonlinear systems. Its numerical value is

calculated on the basis of our measurement from the kurtosis or curvature of the

system. In fact, Lyapunov exponent is the average of the exponential divergence or

convergence rate of chaotic trajectories in state space. Lyapunov exponent measures

the sensitivity of the process to its initial values. Positive Lyapunov exponent

indicates chaotic behavior in the series while its negative value indicates damping

7

dynamic systems (Ellner et al., 1991). Besides, the inverse of maximal Lyapunov

exponent points out the distance between fixed and randomness of the series.

Therefore, based on its numerical value one can determine the predictability degree

(the number of approved predictable periods) of the series (Wolf et al., 1985). This

method was implemented for the purpose of this study.

4.2. Long Memory

Long memory property is a sign of strong correlation between far-distance

observations in a given time series. Hurst (1951), first, noticed that some time series

have this property. However, in mid 1980s, after suggestion of critical concepts like

unit root and cointegration, econometricians realized some other types of

nonstationarity and partial stationarities which are frequently found in economic and

financial time series (Granger and Joyeux, 1980).

Generally, econometricians, used first-order differencing in their empirical

analyses due to its ease of use (in order to avoid the problems of

spurious regression in non-stationary data and the difficulty of fractional

differencing). Undoubtedly, this replacement (of first-order differencing

with fractional differencing) leads to over -or under- differencing and

consequently loss of some of the information in the time series (Huang,

2010). On the other hand, considering the fact that majority of the

financial and economic time series are non-stationary and of the

Differencing Stationary Process (DSP1) kind, in order to eliminate the

problems related to over differencing and to obtain stationary data and

get rid of the problems related to spurious regression, we can use

Fractional Integration.

4.2. Diagnosing Long Memory Process Diagnosing the long memory process is the most important step. Auto Correlation

Functions (ACF) as a graphical test and spectral density test or Geweke and Porter-

1 And some are also trend stationary processes

8

Hudak (GPH) test as the frequently used numeric tests are two main groups of tests

that diagnose the long memory feature.

4.3. Conditional Heteroskedasticity Models These models were introduced by Engle (1982) and elaborated by Bollerslev (1986).

After that, several other conditional heteroskedasticity models were suggested (see

Arouri et al., 2010a). In this study, the Fractional Integrated Generalized

Autoregressive Conditional Heteroskedasticity (FIGARCH) model was used. This

model was first suggested by Baillie et al. (1996). FIGARCH, in general, specifies as

tt

dLBLL )()()1( 2 in which )(L and )(LB are two lag functions with the

optimal lag length of q and p, respectively; L is the lag operator and d is the

fractional parameter. When 0d FIGARCH reduces to a usual GARCH model and

when 1d , FIGARCH is equal to IGARCH model (see Arouri et al., 2010a). In

contrast with durable shocks in IGARCH or transient shocks in GARCH, in

FIGARCH models it is assumed that imposed shocks have moderate durability and

damp at a hyperbolic rate.

4.4. Wavelet Decomposition Wavelet technique, using base functions, transmits time series to the frequency space

and decomposes it in various scales. Contrary to fourier transformation which is only

based on sine functions, wavelet decomposition includes various discrete and

continuous base function, albeit all have finite energy (Reis et al., 2009). This

property makes wavelet a pleasant tool for analyzing nonstationary and transient

series. Such decompositions can be classified into two main categories; Continuous

Wavelet Transform (CWT) and Discrete Wavelet Transform (DWT) (Karim et al.,

2011). Crude oil price series is a discrete series. The most important DWT base

functions include Haar, Daubechies, Symmelets, Coiflets, and Meyer functions.

Among others, Daubechies function is widely used discrete base function (Al Wadi

and Ismail, 2011); considering the similarity between the time series of Iran’s crude

oil price and a specific Daubechies base transformation function called db3, we used

it for decomposing oil price series.

9

5. Empirical Results

5.1. Data In this study, the daily data related to Iran’s heavy crude oil price from 2/1/2002 to

11/3/2011 were used. Table 1 reports the main descriptive statistics for the series of

natural logarithm of oil price (LOIL) as well as oil price return series (DLOIL).

Table 1: Descriptive Statistics

Return Of Oil Prices

Series Tests

Return Of Oil Prices

Series Stat.

-47.572(0.000) ADF1 2536 Observations -47.659(0.000) PP2 0.000677 Mean

0.0355(3. 26) ERS3 0.021500 S.D

23.083(0.010) Box- Ljung Q(10) -0.291624 Skewness

477.43(0.000) McLeod-Li Q2(10) 6.187903 Kurtosis

25.209(0.000) ARCH

10)=F(10,2514) 1109.807(0.000) Jarque- Bra

* All of numbers in parenthesis are probability of related test, but ERS test except that the critical value of the test is.

Source: Findings of Study

As seen in the table 1, the return series of crude oil price has the mean of 0.000667

and standard deviation of 0.0215 in the sample period suggesting that it has been

highly volatile. Besides, Jarque-Bera and kurtosis statistics show that the series not

only is not normally distributed but has wide tails. Based on the Ljung-Box statistics

(10 lags), the null hypothesis of “No serial correlation” is rejected. Similarly,

McLeod-Lee statistics reject the null hypothesis of “No serial correlation in squared

series” and confirm Heteroskedasticity in return series suggesting that there exists

some sort of nonlinear relationship in the squared series. This conclusion is also

approved by Engle’s ARCH test. Finally, according to unit root tests _ADF and PP

tests_ the return series is stationary but ERS test unit root test shows this series is non-

stationary. Thus, such conditions might have been caused by the long memory feature

in this series. For this reason, tests for checking the existence of this feature will be

focused upon in the next part.

1 Augmented Dicky-fuller Test 2 Phillips-Perron Test 3 Elliott-Rothenberg-Stock Test

10

5.2. Predictability of Oil Price

i. Variance Ratio Based on Lo & MacKinlay (1988), the variance ratio test investigates the Martingale

hypothesis.

Table 2: Variance Ratio Test

Criterion Prob. d.f Value

Variance ratio test 0.000 2535 15.99

Source: Findings of Study

As shown in Table 2, the martingale hypothesis –in the return series and its lag

series- is strongly rejected. So, it can be concluded that the generating process of the

data is not random walk; i.e. the series is predictable.

ii. BDS Test This test was developed by Brock, Dechert and Scheinkman (1987). The main

concept behind the BDS test is the correlation integral, which is a measure of the

frequency with which temporal patterns are repeated in the data. BDS test makes it

possible to distinguish between a nonlinear and a chaotic process. The results of

BDS test are presented in Table 3.

Table 2: BDS Test Prob. z-Stat. Std. Error BDS Stat. Dimension

0.0000 7.667829 0.001635 0.012537 2 0.0000 10.53126 0.002591 0.027289 3 0.0000 12.32419 0.003077 0.037924 4 0.0000 13.69544 0.003198 0.043804 5

Source: Findings of Study

As seen in Table 2, the null hypothesis of “the residual series is not random” is

rejected. This result approves the existence of a nonlinear (may be a chaotic) process

in the data. It should be noted that when BDS test in 2 (or higher) dimensions rejects

the hypothesis that the series is random; existence of a nonlinear process is quite

probable. This result points to the conclusion that BDS test also approves that the

data generating process in this study is nonlinear.

11

iii. Maximal Lyapunov Exponent Maximal Lyapunov Exponent measures the rate of convergence (divergence) of two

initially close points along their trajectory over time. In Lyapunov method, this rate

is measured by an exponential function. The value of these exponents can be used to

investigate the Local Stability of linear or nonlinear systems. In this test, positive

values of exponents indicate exponential divergence of the series, high sensitivity to

initial conditions and therefore, chaotic process. On the other hand, negative values

of exponents indicate exponential convergence. Finally, when Lyapunov exponents

are zero, one may argue that there is no converging or diverging trajectory in the

data; i.e. the series follows a fixed process. Therefore, in most of the cases, existence

of only one Lyapunov exponent is enough to conclude that the system is chaotic.

There are two main approaches for computing Maximal Lyapunov Exponent; (1)

direct method; and (2) Jacobian Method. In this study, we implemented both methods

(using MATLAB software); the results are reported in Tables 4 and 5.

Table 4: Direct Approach to Maximal Lyapunov Exponent

Max Lag 1 2 3 4 5

Lyapanov Exponential 0.0743 0.0489 0.1075 0.1097 0.0956

Maximal Lyapunov Exponent 0.1097

Source: Findings of Study

Table 5: Jacobian Approach to maximal Lyapunov exponent

Max Lag 1 2 3 4 5

Lyapanov Exponential 0 0 0. 0560 0 0.0537

Maximal Lyapunov Exponent 0.0560

Source: Findings of Study

As seen in Tables 4 and 5, both methods lead to positive Lyapunov exponents.

Besides, due to high sensitivity to initial conditions, for each combination of initial

points, close trajectories diverge quickly in a way that there is no fixed point or

cycle. Then, one may conclude that this process is chaotic. Furthermore, the maximal

Lyapunov exponent in the direct method was 0.1097; so, the predictability limit (the

maximum number of predictable periods) is approximately 10 days. In Jacobian

method, this number was 18 days. To be prudent, we have selected the minimum

value (10 days) for the rest of calculations and estimations. Therefore, achieving

this result can not only increase the validity and strength of out-of-sampling

forecasting but decrease the number of forecasting errors as well.

12

5.3. Quantitative analysis of the Long Memory Process Estimating the long memory parameter (d) is the milestone of modeling long

memory property. ACF and GPH are two commonly used methods for this purpose.

Graph 1 depicts the ACF of the logarithm of the time series of crude oil price. As

clearly shown, following an exponential trend, graph decreases very smoothly, a

typical shape for time series that are non-stationary and have the long memory

property.

Graph 1: ACF of LOIL

Source: Findings of Study

If such a series does not have the long memory property, it is expected that after first

differencing, the series would become stationary. According to Table 6, although

ADF and PP tests recognize the oil price series stationary after first differencing, the

ERS and KPSS tests show some sort of non-stationarity in the data. This result

further indicates the existence of the long memory property.

Table 6: Unit Root Tests Result Critical

Value Accounting

Value Tests

Stationary -1.9409 -47.572 ADF

Stationary -1.9409 -47.659 Phillips-Perron

Non-Stationary 3. 26 0.0355 ERS

Non-Stationary 0.463 2.159 KPSS

Source: Findings of Study

Models considering long memory property are very sensitive to the estimation of

long memory parameter as well as the pattern of damping of auto-correlation

Lag

ACF

13

functions. In this study, the long memory parameter was estimated using GPH

approach. This method, invented by Gewek, Porter-Hudak (1987), is based on

frequency domain analysis. GPH method applies a special regression technique

called Log-Period Gram which allows us to distinguish between long-term and short-

term trends. The slope of regression line calculated by this technique is exactly equal

to long memory parameter.

Table 7 reports the estimated long memory parameter for both the logarithmic series

and return series. To do so, we have used OX-Metrics software.

Table 7: Estimated Long Memory Parameters

Prob. t-stat. d-

Parameter Variable

0.000 65.8 1.05207 LOIL

0.005 2.79 0.440238 dLOIL

Source: Findings of Study

As table 7 shows, the estimated long memory parameter is statistically significant,

i.e., it is not equal to zero suggesting that the series of (logarithm of) crude oil price

in the level has the long memory property. However, the return series should be

modeled after another differencing.

5.4. Modeling the Return Series of Crude Oil Price Knowing that the crude oil price in the level has the long memory property, in this

step, we fit an econometric model to our data. In this paper, the most famous and

flexible long memory model, i.e., ARFIMA was applied to specify the mean

equation:

(3 ) TtLyLL ttt

d ,...,3,2,1)()()1)((

)(L and )(L indicate AutoRegressive (AR) and Moving Average (MA)

polynomials, respectively. L is the lag operator and t represents the mean of the

series. d is the differencing parameter and dL)1( stands for fractional differencing

operator. If d=1, this model reduces to ARIMA model. If, on the other hand, 5.0d ,

the covariance is fixed and if 0d , long memory property exists (Husking, 1981).

When 5.00 d , ACF has a hyperbolic decreasing pattern and when 05.0 d ,

medium-term (or short-term) memory exists; this property suggests that too many

14

differencing have been made. In such cases, the invert of ACF has a hyperbolic

decreasing pattern.

To estimate the ARFIMA model (and d parameter), three methods were

implemented; Exact Maximum Likelihood (EML); Modified Profile Likelihood

(MPL); and Non-Linear Least Square (NLS). Table 8 compares various estimated

models on the basis of Akaike Information Criterion (AIC). As shown in this table,

ARFIMA (1, 0.14, 2) has the best performance compared to other models.

Table 8: Estimated ARFIMA models

ARCH-TEST AIC

Model EML NLS MPL

F(1,2519)=31.482(0.000) -4.85611325 -4.85645431 -4.84134185 ARFIMA(1,0.04,1)

F(1,2518)= 34.235(0.000) -4.85659337 -4.86294341 -4.84141198 ARFIMA(1,0.04,2)

F(1,2518)= 33.354(0.000) -4.85668659 -4.85703734 -4.8415002 ARFIMA(2,0.04,1)

F(1,2517)=32.825(0.000) -4.85928044 -4.85701168 -4.84418547 ARFIMA(2,0.04,2)

Source: Findings of Study

Moreover, with respect to volatility equation, diagnostic ARCH tests approved the

existence of ARCH effects in the residual series; to model this conditional

heteroskedasticity, fractional (to track the long memory property) and non-fractional

GARCH models were estimated. Table 9 compares them on the basis of AIC and

Schwarz-Bayesian Criterion (SBC).

Table 9: ARFIMA-FIGARCH Models

ARFIMA(2,2) ARFIMA(2,1) ARFIMA(1,2) ARFIMA(1,1) Models

AIC SBC AIC SBC AIC SBC AIC SBC

-

5.5486

-

5.4535

-

5.5459

-

5.4571

-

5.5477

-

5.4589

-

5.5492 -5.4439 GARCH

-

5.5242

-

5.4164

-

5.5182

-

5.4168

-

5.5186

-

5.4172

-

5.5116 -5.4165 EGARCH

-

5.5509

-

5.4495

-

5.5481 -5.453

-

5.5497

-

5.4546

-

5.5512 -5.4624 GJR-GARCH

-

5.5507

-

5.4429

-

5.5487

-

5.4472

-

5.5505

-

5.4491

-

5.5391 -5.4575 APGARCH

-

5.5391

-

5.4504

-

5.5363

-

5.4538

-

5.5383

-

5.4558

-

5.5398 -5.4637 IGARCH

-

5.5399

-

5.4385

-

5.5371 -5.442

-

5.5526

-

5.4668

-

5.5407 -5.452

FIGARCH(BBM

)

-

5.5399

-

5.4385

-

5.5371

-

5.4421

-

5.5391 -5.444

-

5.5408 -5.452

FIGARCH

(Chang)

Source: Findings of Study

15

Table 9 has three different parts: part 1 includes non-fractional heteroskedasticity

models; part 2 is dedicated to some integrated non-fractional heteroskedasticity

(IGARCH) models; and part 3 includes various fractional heteroskedasticity

(FIGARCH) models. According to both information criteria, ARFIMA(1,2)-

FIGARCH(BBM) proves to be the best specification. Table 10 reports the

coefficients of this model as well as some diagnostic statistics in detail. As the table

clearly shows, all the coefficients are significant (in 95% of confidence). Ljung-Box

statistics shows no sign of serial correlation between residuals. Besides, according to

both McLeod-Lee and ARCH statistics, there is no heteroskedasticity.

Table 10: Estimation of ARFIMA(1,2)-FIGARCH(BBM) Model

Prob t-Stat. Standard Error Coefficient Variable

Mean Equation

0.0055 2.779 0.00036412 0.001012 C

0.0000 12.81 0.0034389 0.044040 d-ARFIMA

0.0184 -2.359 0.37596 -0.887022 AR(1)

0.0076 2.671 0.35649 0.952358 MA(1)

0.0082 2.570 0.023550 0.060525 MA(2)

0.0000 19.74 0.0071753 0.141621 Dum

Variance Equation

0.0992 1.649 0.037146 0.061262 C

0.0000 3.599 0.11706 0.421338 d-FIGARCH

0.0000 5.474 0.052668 0.288289 ARCH

0.0000 8.089 0.089571 0.724586 GARCH

10.3996(0.167) Box- Ljung Q(10) 6365.271 Log likelihood

5.80142(0..669) McLeod-Li Q2(10) -5.552605 Akaike

0.56651(0.842) ARCH(10)=F(10,2503) -5.466807 Schwarz

Source: Findings of Study

5.5. Wavelet Decomposition To determine the optimized decomposition level, the series was first decomposed up

to 5 levels and then using graphical wavelet toolbox in MATLAB software, the best

level was determined, which is one. It was also found that one of the Daubechies

functions called db3 fits the data better than other base functions. Graph 2, depicts

the approximation -A(1)- and detailed -d(1)- series resulting from decomposing oil

price series with db3 up to 1 level.

16

Graph 2: Decomposed Series of Oil Price up to 1 Level Using db3

Source: Findings of Study

Modeling oil price fluctuations using approximation and detail series yielded

valuable results. According to the results, the long memory property significantly

increased in the “detail” series while not in the “approximation” series. Furthermore,

comparing the accuracy of forecasts based on decomposed (DCM) and non-

decomposed (NDCM) data are reported in Table 11.

Table 11: Forecasting Performance of Models Based on DCM and NDCM Data

Decomposition data

RMSE MSE Model

0.00525 0.0000276 ARFIMA-FIGARCH

Non-Decomposition data

RMSE MSE Model

0.02298 0.0005281 ARFIMA-FIGARCH

Source: Findings of Study

In line with previous studies mentioned in the literature (e.g., He et al., 2012; Haidar

& Wolff, 2011, to name a few), DCM data have led to significantly more accurate

forecasts as seen in the Table 11.

17

6. Conclusion The strategic role of crude oil price and its deep wide effects on all countries in the

world including Iran instigated the conduction of numerous studies in recent

decades. Following this trend, we compared the performance of some various models

in forecasting the crude oil price; these models include fractal chaos-based models,

different GARCH-type volatility models and models based on DCM data using

wavelet decomposition technique.

Our findings suggest that crude oil price series is not Martingale (i.e., it is

predictable) and follows a nonlinear trend. So, the Efficient Market Hypothesis

(EMH) is not valid in this case. Moreover, the results approved the Fractal Market

Hypothesis (FMH) in the oil market; so, it can be concluded that this series is

chaotic. Using -invert of- Maximal Lyapunov Exponent the predictability

limit was determined as 10 days.

On the other hand, due to confirmed ARCH effects, GARCH-Type various models

were used to track the variance fluctuations more accurately and reduce the

forecasting error. Besides, the existence of long memory property in first and second

momentums of the series (approved by ACF and GPH tests) led the researchers to fit

an ARFIMA-FIGARCH specification model to crude oil price data. Compared with

various fractal and non-fractal model, the nonlinear ARFIMA (1,2)-FIGARCH

(BBM) model had the best forecasting performance on the basis of AIC and SBC

information criteria. It is worth mentioning that, there are some complicated models

which may improve forecasting performance in special cases even more than the

model used in this study by including the long memory property in the econometric

model which leads to systematic improvement.

Another goal of this paper was to compare the forecasting accuracy of models using

the wavelet decomposed data with the models that use the original data. Our findings

approve that the smoothed data are likely to provide more accurate forecasts not only

in first momentum (mean) but also in second momentum (variance) of the crude oil

price series.

In brief, it can be concluded that considering the long memory property and applying

hybrid methods lead to more accurate forecasts. Besides, the best model in this study

(ARFIMA-FIGARCH model using DCM data) can be used in forecasting other

18

market indicators (e.g., Value at Risk (VaR)) in oil or other highly volatile financial

markets.

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